Tuning with Just Intonation and Equal Temparement

Tuning With Just Intonation and Equal Temparement

Jacob Scott December 9, 2015

1 A Brief Introduction

The discovery that the freqency of a vibrating string is inversely proportional to its length is generally credited to the Greek polymath Pythagoras in the late 6th century BC [1]. The implication of this discovery was that for any string of length l vibrating at frequency λ hertz, a 1 string of length 2 l will vibrate at 2λ hertz. Doubling the frequency in this way (which can be done by artiﬁcially creating a node exactly half way along the string) gives the ﬁrst harmonic in the harmonic 1 series. The second harmonic would be a string of length 3 l vibrating at 3λ hertz, and so on [4]

(Note: from this point the stan- dard ratio/fraction systems will be fundamental frequency used to denote pitches in just intonation. For more information, see [5].) Pythagoras found that by repeatedly 1st harmonic (octave) taking the interval between the 1st and 2nd harmonics (i.e. a perfect fifth), all 12 notes of the chromatic scale could be found (for more ex- 2nd harmonic (Perfect/Pythagorian fifth) planation see the cycle of fifths [2]). However, completing this cycle using ’true’ harmonics, the final frequency 3rd harmonic (second octave) of the cycle was found to be ’off’ from the original, and to compensate, a highly dissonant ’wolf fifth’ had to be used to complete the oc- 4th harmonic (major third in second octave) tave. (See https://www.youtube. com/watch?v=Dl7iIzvUMGg). Many different methods of tuning have been invented to try and solve (for sound, see https://soundcloud.com/jakescottmusic/harmonics) the problem of this apparent error in the fundamental mechanics of music. Figure 1: The first four harmonics Broadly, they can be categorised un- der equal temperament and systems of just intonation.

2 Equal Temperament

The modern system of tuning simply divides the octave into 12 equal steps. A formula for this can be derived as follows: If Pn is a known frequency and Pa is exactly one octave higher than Pn, then Pn = 2Pa. 12 Since diﬀerences in frequency√ are not additive but multiplicative, Pn = Pax Therefore x12 = 2 =⇒ x = 12 2

1 √ 12 12 And from this, Pn = Pa( 2) √ 12 n If the frequency of Pa is known, then Pn can be found using the formula Pn = Pa( 2) . Moreover, if Pn and Pa are n and a semitones from some other ’base’ frequency, respectively, then we can use the following formula: √ 12 n−a Pn = Pa( 2)

2.1 Cents 1 The cent is a logarithmic unit in music theory, denoting 100 th of a semitone. The advantage of using cents is that they can be used additively to ﬁnd the diﬀerence between two frequencies, which gives a much clearer and more intuitive representation of small intervals. If we take the previous formula, with a = 0, we can derive the formula for the cent. √ 12 n Pn n Pn n Pn Pn = Pa( 2) =⇒ = 2 12 =⇒ log2( ) = log22 =⇒ n = 12log2( ) P0 P0 12 P0 Multiplying by 100 to give cents we obtain:

Pn c = 1200log2( ) P0 3 Just Intonation

Just intonation is any system of tuning where intervals are given by ratios of ’small’ positive integers. Many systems of just intonation have been devised that fit this definition, but for the purposes of this project I will be using five-limit just intonation, and particularly assymetric five-limit just intonation. For more information on types of just intonation, see [5] and [3].

3.1 Five-limit Just Intonation (Note: the following can be found in greater detail online, see [3])

The fractions in Table 1 are found as follows: Factor 1/9 1/3 1 3 9 For any given note, the corresponding fac- 5 Note D− A E B F]+ tors are multiplied together to give the the Ratio 10/9 5/3 5/4 15/8 45/32 correct fraction for that note at some oc- 1 Note B[ F C G D tave. To obtain the note in its correct oc- Ratio 16/9 4/3 1 3/2 9/8 tave, the fraction is multiplied by 2 to a 1/5 Note G[− D[− A[ E[ B[ chosen integer power. Ratio 64/45 16/15 8/5 6/5 9/5 1 1 6 In the case of E[, 5 · 3 · 2 = 5 . In Table 1, notes with enharmonic equiva- Table 1: Fifteen Pitches lents (i.e, two notes that whilst occupying the same position on the keyboard, do not have the same exact frequency) have been shaded with matching colours. To obtain ﬁve-limit tuning, the shaded frequencies on the left of the table (i.e. the notes corresponding to fractions 10/9, 16/9 and 64/45) are discarded, giving a full 12-tone justly intonated scale. (Note: there are other systems for eliminating enharmonic equivalents, this one in particular gives asymmetric ﬁve-limit tuning.)

Sagemath can be used to create a graph showing the diﬀerences in cents between equal-tempered and justly intonated pitches, as shown in Figure 2. The complete code used to create the graph can be found at https://cloud.sagemath.com/projects/bd250a4b-64e4-4530-aa25-0d15b6d41b61/ files/pitch%20graph.sagews

2 (for an aural comparison see https://soundcloud.com/jakescottmusic/equal-temperament-just-intonation)

Figure 2: Diﬀerence in cents between justly intonated and equal-tempered pitches

4 Tuning in Python

The following code shows how python can be used to calculate equal-tempered and justly intonated frequencies, and the difference between them in cents. The program the code is taken from has the primary purpose of using randomly generated chords to highlight the differences in these tuning systems, though it also offers the ability to tune chords inputted by a user. The chord Cm7 (made up of notes C, E[, G and B[) in the key of C particularly demonstrates the difference between just intonation and equal temperament. (See Figure 2: E[ and B[ correspond to semitones 3 and 10 respectively - points at which the blue line moves particularly far from the red.) The full code can be found at https://cloud.sagemath.com/ projects/bd250a4b-64e4-4530-aa25-0d15b6d41b61/files/tuner%20final%20program.sagews, but note that the python module ’pyo’ must be installed in order to generate audio using the program. freqs_equal = [] # creates an empty list for the final equal-tempered frequencies to be added to for note in notelist: letter = note[0] # separates ’note’ element into its octave and the note’s name. octave = note[1] freq = 440 * ((2 ** (1.0/12.0)) ** (letter - 9 - (12 * 4))) # calculates note in its lowest octave, relative to the frequence of A4 (440 Hz) if octave != 0: freq = freq * ((2 ** (1.0/12.0)) ** (octave * 12)) # transposes the note to its correct octave freq = int(round(freq, 0)) # rounds up freq and turns it into an int freqs_equal.append(freq) # adds the found frequency to the list self.freqs_equal = freqs_equal freqs_just = [] for note in notelist: letter = note[0] octave = note[1] # get the note’s name and octave if letter - key >= 0: interval = letter - key # calculates interval else: interval = (letter + 12) - key octave -= 1 # calculates interval for a different octave, changes octave to compensate cents = [] #creates an empty list for e in range(0, len(notelist)): #do this as many times as there are notes in the chord justfreq = freqs_just[e] equalfreq = freqs_equal[e] difference = 1200 * math.log((float(justfreq) / float(equalfreq)), 2) # uses formula to find difference in cents between two frequencies if difference < 0:

3 difference = 1200 * math.log((float(equalfreq) / float(justfreq)), 2) # if the difference is negative, do it the other way round difference = int(round(difference, 0)) # rounds up difference and turns it into an int cents.append(difference) # add the difference to the list self.cents = cents

5 Conclusions

In this report I have briefly explained the difference between equal-temperament and just intonation, giving examples of how the differences can be represented both visually and aurally using sagemath and python. This project could be developed further by using a specifically music-orientated programming language such as ChucK to investigate how the tuning capabilities of computers can be used practically in areas such as composition and performance. In addition to this, many non-Western musical cultures do not split the octave into 12 notes as we do. Exploring how the harmonic series relates to tuning systems such as these could potentially offer much more insight into the mathematics of tuning.

References

[1] John H Chalmers. Divisions of the tetrachord: a prolegomenon to the construction of musical scales. Frog Peak Music, 1993.

[2] Wikipedia. Circle of ﬁfths — wikipedia, the free encyclopedia, 2015. [Online; accessed 9-December- 2015]. [3] Wikipedia. Five-limit tuning — wikipedia, the free encyclopedia, 2015. [Online; accessed 6-December- 2015]. [4] Wikipedia. Harmonic series (music) — wikipedia, the free encyclopedia, 2015. [Online; accessed 5-December-2015]. [5] Wikipedia. Just intonation — wikipedia, the free encyclopedia, 2015. [Online; accessed 5-December- 2015].